Determination of the S-wave scattering shape parameter P from the zero-energy wave function

We show that for S-wave scattering at an energy k2 by a local potential which supports no more than one bound state, the shape parameter P and coefficients of higher powers of k2 in the effective range expansion function cotδ=-1/a+1/2r0k2-Pr03k3+Qr05k6+..., where δ is the phase shift, may be obtained from the zero-energy wave function, u0(r). Thus δ itself may be determined from u0. We show that Pr03=F0R[β(r)u02(r)-β¯(r)u¯ 02(r)]dr, where r0 is the effective range, β(r) is determined from an integral involving the wave function, and β¯(r) is a simple function of r which involves the scattering length and effective range.